The article 入瞳 introduces the entrance pupil in paraxial imaging, assuming that the entrance pupil and the aperture stop are a pair of conjugate object and image. This one-to-one concept of the entrance pupil applies to ideal lenses or telephoto (long focal length, narrow angle) lenses. However, if we observe a fisheye (ultra-wide-angle) lens, we will find that the position of its entrance pupil is not fixed, but moves as the viewing position changes, tracking the observer like an eye. In this article, we will explore the mechanism behind this phenomenon and its impact on camera image quality measurement.

Effect of Fisheye Lens Structure on the Entrance Pupil

Figure 2 shows an example of a fisheye lens structure. We can see that there are two large-diameter lenses in front of the aperture stop, both of which are negative meniscus lenses that are convex on the front, concave on the back, and thin in the middle. Such a front group can deflect the extremely oblique rays from the edge of the object field towards the optical axis to pass through the aperture stop. This is the fundamental reason why fisheye lenses can achieve a wide field of view. However, such a structure also challenges the concept of the entrance pupil.

The concept of the entrance pupil is the image of the aperture stop formed by the lens group in front of it. According to this concept, we can try to find the entrance pupil of a fisheye lens. As shown in Figure 3, we assume the object is at infinity (i.e., the incident light is parallel light), and select three object chief ray angles of 0°, 45°, and 90°. By performing ray tracing in the meridional plane, we can obtain three entrance pupils at different positions (indicated by red short lines). The entrance pupil at 0° is consistent with the concept; its center is located on the optical axis, the plane where the entrance pupil is located is perpendicular to the optical axis, and it is located behind the second lens. However, as the chief ray angle increases, the entrance pupil begins to tilt. This phenomenon is consistent with that shown in Figure 1, where the entrance pupil tracks the observer like the pupil of an eye. In fact, only when the entrance pupil tilts following the tilt of the chief ray can the incident light from the scene pass through the aperture stop. Imagine if the entrance pupil were always perpendicular to the optical axis; then, for incident light at 90°, the projected area of the entrance pupil would be 0, making it impossible to pass through the aperture stop. Therefore, from the perspective of radiometric transfer, the tilt of the entrance pupil is inevitable for a fisheye lens.

In addition to tilting, the center of the entrance pupil also moves forward as the object chief ray angle increases. The blue dashed line in Figure 4 shows the trajectory of the entrance pupil center in the meridional plane. For an actual fisheye lens, the set of entrance pupil centers lies on a curved surface formed by rotating this curve around the optical axis. In fact, the forward movement of the entrance pupil is also inevitable. If we compare the lens to a well, the first lens is like the wellhead. If the entrance pupil is “hidden” at the bottom of the well, it would be like a frog in a well looking at the sky. To increase the field of view, it is necessary to raise the position of the entrance pupil to near the wellhead, or even above it (in front of the first lens), in order to achieve a field of view greater than 180°.

Entrance Pupil Center, Field of View, and No-Parallax Point
For an ideal lens, the chief rays of different fields converge at the center of the entrance pupil. In three-dimensional space, when the chief ray angle is constant, the chief rays in all meridional planes will form the lateral surface of a right circular cone. The apex of the cone is the center of the entrance pupil, located on the optical axis, and half of the cone angle is the chief ray angle. Different chief ray angles correspond to different cones, and all cones share the same apex. Obviously, the definitions of field and field of view angle are both based on this apex. For a fisheye lens, when the chief ray angle is constant, the chief rays in all meridional planes will still form a right circular cone, and the apex of the cone will still be located on the optical axis, but at this time, the center of the entrance pupil will form a circle and will no longer be located at the apex of the cone (except when the chief ray angle is 0°). When the chief ray angle changes, the apex of the corresponding cone will move accordingly, so all cones no longer share the same apex. However, since the apex of the cone is still the intersection of the extended lines of the chief rays, we can still use this apex to define the field of view, as shown in the left image of Figure 5.

For an ideal lens, assuming an object point located on the optical axis rotates by an angle around the entrance pupil center (cone apex) in any meridional plane, then this object point must lie on the chief ray of this angle, and this angle corresponds to the respective field position. However, for the fisheye lens shown in the figure, since the cone apex is not unique, assuming an on-axis object point rotates by about 67° around the cone apex at 0° (i.e., the entrance pupil center), it actually lies on the object chief ray of about 73° (as shown in the right image of Figure 5). This is the result of the movement of the entrance pupil center. In the actual testing of fisheye lenses and cameras, if the imaging performance at different field positions is to be measured, it should be noted that simply rotating the device under test or the target around a certain cone apex is not enough; it is also necessary to move the target or the device under test along the optical axis to ensure that the center of the target is always on the chief ray of the target field.

In addition, this leads to another interesting concept—the no-parallax point. Since the entrance pupil center of an ideal lens is fixed, if there are two object points A and B in the same field but at different object distances (A is in front of B), A will occlude B. Moreover, as long as the lens or the object points A/B rotate around the entrance pupil center, this occlusion will remain unchanged, or in other words, the relative positional relationship between the foreground and background remains unchanged, as shown in the left image of Figure 6. The entrance pupil center at this time is called the no-parallax point. However, for a fisheye lens, suppose there are two object points A and B located in the 0° field, and A completely occludes B. When the lens or the two object points rotate by a certain angle around the entrance pupil center of the 0° field simultaneously, A and B will be located on chief rays of different angles, so A no longer completely occludes B, or in other words, the relative positional relationship between the foreground and background changes, as shown in the right image of Figure 6. Therefore, the fisheye lens in the figure does not have a no-parallax point. The concept of the no-parallax point is very important for panoramic image stitching. If the camera rotates around the no-parallax point during shooting, the positional relationship between the foreground and background in the image remains unchanged, making image stitching relatively simple. Otherwise, image stitching must attempt to compensate for parallax (changes in the positional relationship between the foreground and background). In addition, for an ideal imaging lens, the entrance pupil center, the no-parallax point, and the origin of the camera coordinate system in geometric calibration coincide, but for a fisheye lens, this assumption usually does not hold.

Having written this far, everyone should now have a preliminary understanding of the concept of the entrance pupil in fisheye lenses. In the actual testing of fisheye cameras and lenses, if the device under test does not have a unique no-parallax point or its position cannot be precisely located, we can start from the phenomenon and try to find the position with the minimum parallax as the equivalent no-parallax point of the device under test, so as to minimize the impact of the position change of the entrance pupil center on imaging and measurement results. Considering that the positions of the equivalent entrance pupil center or equivalent no-parallax point vary for different cameras, when using the Yanding RFT series comprehensive tester, users can finely adjust the position of the camera under test.